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Let $S$ be a set and $\circ$ a binary operation on it. $\circ$ is said to be commutative if
$a\circ b=b\circ a$ 
for all $a,b\in S$.
Viewing $\circ$ as a function from $S\times S$ to $S$, the commutativity of $\circ$ can be notated as
$\circ(a,b)=\circ(b,a).$ 
Some common examples of commutative operations are

multiplication over the integers: $m\cdot n=m\cdot n$ for all integers $m,n$

addition over $n\times n$ matrices, $A+B=B+A$ for all $n\times n$ matrices $A,B$, and

multiplication over the reals: $rs=sr$, for all real numbers $r,s$.
A binary operation that is not commutative is said to be noncommutative. A common example of a noncommutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general,
$ab\neq ba.$ 
For instance, $21=1\neq1=12$.
Other examples of noncommutative binary operations can be found in the attachment below.
Remark. The notion of commutativity can be generalized to $n$ary operations, where $n\geq 2$. An $n$ary operation $f$ on a set $A$ is said to be commutative if
$f(a_{1},a_{2},\ldots,a_{n})=f(a_{{\pi(1)}},a_{{\pi(2)}},\ldots,a_{{\pi(n)}})$ 
for every permutation $\pi$ on $\{1,2,\ldots,n\}$, and for every choice of $n$ elements $a_{i}$ of $A$.
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